Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

Abstract

Let be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space . Let be maximal monotone and quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space , dense and continuously embedded in . Assume, further, that there exists such that d for all and . New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type . A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator is given as a result of surjectivity of , where is of type with respect to . These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in of a nonlinear parabolic problem of the type ,  ; ,  ; ,  , where , is a nonempty, bounded, and open subset of ,    satisfies certain growth conditions, and , , and is the conjugate exponent of .

1. Introduction—Preliminaries

In what follows, is a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space The norm of the space , and any other normed spaces herein, will be denoted by For and , the pairing denotes the value Let and be real Banach spaces. For a multivalued mapping , we define the domain of by and the range of by . We also denote the graph of by A mapping is “demicontinuous” if it is continuous from the strong topology of to the weak topology of . A multivalued mapping is “bounded” if it maps bounded subsets of to bounded subsets of . It is “compact” if it is strongly continuous and maps bounded subsets of to relatively compact subset of It is “finitely continuous” if it is upper semicontinuous from each finite dimensional subspace of to the weak topology of . It is “quasibounded” if for every there exists such that with and imply It is “strongly quasibounded” if for every there exists such that with and imply In what follows, a mapping will be called “continuous” if it is strongly continuous.

Let be continuous strictly increasing function such that and as The duality mapping corresponding to denoted by is defined by It is well-known that, for each , the Hahn-Banach Theorem implies . Since and are locally uniformly convex, is single-valued, bounded monotone of type and bicontinuous. If for , then is denoted by and is called the normalized duality mapping.

An operator is said to be “monotone” if, for every , , and every , , we have A monotone mapping is “maximal monotone” if for every ; that is, is maximal monotone if and only if is monotone and for every implies and If is maximal monotone, the operator , , defined by , is bounded, continuous, maximal monotone and such that as , for every , where The “resolvent” , defined by , is continuous and for every Moreover, for all , where is the convex hull of the set An operator is called “coercive” if either is bounded or there exists a function such that as and for all and For an operator and , we denote It is called weakly coercive if either is bounded or as .

The following definitions are used throughout the paper. In arbitrary Banach space , Browder and Hess [1] introduced the definitions of pseudomonotone and generalized pseudomonotone operators. The original definition for single-valued pseudomonotone, generalized pseudomonotone, and operators of type with domain all of , is due to Brézis [2].

Definition 1. An operator is called (i)“generalized pseudomonotone” if, for each sequence in with and as such that , then , , and as (ii)“type ” if, for each sequence in with in and as such that , then and (iii)“-expansive” if there exists such that for all , , , and . It is called expansive if .

We notice here that the definition of single-valued expansive mapping is due to Nirenberg [3]. In order to enlarge the class of single-valued operators, the multivalued version is introduced in (iii) of Definition 1. It is not hard to notice that every uniformly monotone operator is expansive. Furthermore, in a Hilbert space , if is monotone, we see that, for each , is multivalued expansive with domain .

The following definition gives a larger class of operators of monotone type, which can be found in Kartsatos and Skrypnik [4].

Definition 2. Let be maximal monotone and Let be a linear subspace of Then is said to be (i)“quasibounded with respect to ” if, for each , there exists such that where and , then ,(ii)“generalized with respect to ” if, for each in with , in and in as such that for all , then and ,(iii)“generalized pseudomonotone with respect to ” if, for each in with , in and in as such that for all , then , , and as ,(iv)“of type with respect to ” if, for each in with , in and in as such that for all , then and

By Definition 2, it is not difficult to see that and is quasibounded implying that is quasibounded with respect to . Furthermore, it follows that the class of generalized operators with respect to includes the class of operators of type .

For basic definitions and further properties of mappings of monotone type, the reader is referred to Barbu [5], Brèzis et al. [6], Brèzis [2], Browder and Hess [1], Pascali and Sburlan [7], Browder [8], and Zeidler [9]. For results concerning perturbations of maximal monotone operators by bounded and everywhere defined pseudomonotone type operators, the reader is referred to Browder and Hess [1], Brèzis [2], Browder [10], Brèzis and Nirenberg [11], Kenmochi [12–14], Guan et al. [15], Le [16], Guan and Kartsatos [15, 17], and Kartsatos and Skrypnik [4] and the references therein. For recent degree theory and applications for solvability of operator inclusions involving bounded pseudomonotone perturbations of maximal monotone operators under general coercivity and Leray-Schauder type boundary conditions, we cite the paper due to Asfaw and Kartsatos [18]. Existence results concerning noncoercive operators of the type , where is maximal monotone and is bounded pseudomonotone, can be found in the paper due to Asfaw [19]. For applications of the theory of perturbed monotone type operators to variational and hemivariational inequality problems, the reader is referred to the papers due to Carl and Le [20], Carl et al. [21], Carl [22], and Carl and Motreanu [23] and the references therein. For a separable reflexive Banach space and a nonempty, closed, and convex subset of , Asfaw and Kartsatos [24] gave existence results for locally defined operators of the type , where is maximal monotone and is demicontinuous and generalized pseudomonotone under coercivity condition on .

The main contribution of the paper is to obtain surjectivity results for noncoercive and not everywhere defined operators of the type(i), where is quasibounded demicontinuous generalized pseudomonotone such that(a)there exists a real reflexive separable Banach space , dense and continuously embedded in ;(b)there exists such that for all and ;(c)there exist and such that for all and ,(ii), where is quasibounded demicontinuous of type with such that (b) and (c) of (i) are satisfied.

In Section 2, we proved surjectivity results for and satisfying conditions (i) and (ii), respectively. In Theorem 6, we provide a surjectivity result for operators of the type , where and satisfy condition (i). Theorem 6 is new and improves the existing surjectivity results for an operator , which is single-valued, everywhere defined, bounded, and coercive pseudomonotone. In particular, for a single-valued pseudomonotone operator , Theorem 6 improves the surjectivity results due to Browder and Hess [1], Kenmochi [12–14], Le [16], Guan and Kartsatos [17], Asfaw and Kartsatos [18], and Asfaw [19, 25] because the results in these references require to be everywhere defined, bounded, and coercive while Theorem 6 used to be densely defined, quasibounded, and noncoercive. Moreover, Browder (cf. Zeidler [9, Theorem 32. A, pages 866–872]) gave the main theorem for perturbations of maximal monotone operator by a single-valued, bounded, demicontinuous, and coercive operator with , a nonempty, closed, and convex subset of . In view of this, Theorem 6 gives an analogous result, where is dense in , possibly, neither closed nor convex, and is weakly coercive. It is also known, due to Browder and Hess [1], that every pseudomonotone operator from into with is generalized pseudomonotone. It is also true that is demicontinuous provided that it is bounded, single-valued, and everywhere defined. Consequently, the arguments used in the proof of Theorem 6 give analogous conclusion if is bounded pseudomonotone and and satisfy the given hypotheses. As a consequence of Corollary 7, a partial positive answer for Nirenberg’s problem on surjectivity of densely defined demicontinuous generalized pseudomonotone expansive mapping is provided. In addition, Theorem 8 provides surjectivity result for operators of the type , where and satisfy condition (ii). As a result of Theorem 8, a new characterization of linear maximal monotone operator is proved when the space is separable. It is well known due to Brézis (cf. Zeidler [9, Theorem 32. L, pages 897–899]) that a linear monotone operator is maximal monotone if and only if is closed and densely defined and the adjoint operator is monotone. An interesting result in the present paper is that a linear monotone operator is maximal monotone if and only if is closed and densely defined, provided that is separable. This result weakens the monotonicity condition on used by Brézis (cf. Zeidler [9, Theorem . L, pages 897–899]). To the best of the author’s knowledge, Theorem 8 is a new result and Corollary 9 improves the well-known result of Brézis. In Section 3, we demonstrate the applicability of the results by proving existence of weak solution in of a nonlinear parabolic problem, where and is a nonempty, bounded, and open subset of .

The following important lemma is due to Brèzis et al. [6].

Lemma 3. Let be a maximal monotone set in . If such that , as , and either or then and as

Browder and Ton [26] gave the following important embedding result.

Lemma 4. Let be a separable reflexive Banach space. Then there exists a real separable Hilbert space and a compact injection such that

In this paper, we use the following fixed point result for compact operators, originally due to Leray and Schauder, which may be found in the book of Granas and Dugundji [27, Theorem 5.2, page 123].

Lemma 5. Let be a convex subset of a normed linear space and let be nonempty relatively open in with . Then each compact map satisfies that either (i) has a fixed point in or (ii)there exist and such that , where is the boundary of with respect to the subspace topology on

2. Main Results

In this section, we prove the following new surjectivity result for maximal monotone perturbation of densely defined noncoercive generalized pseudomonotone operator in separable reflexive Banach spaces.

Theorem 6. Let be maximal monotone with and quasibounded demicontinuous generalized pseudomonotone. Suppose is a real reflexive separable Banach space dense and continuously embedded in Assume, further, that there exist , , and satisfying for all and either (i)  or (ii)there exists such that as and Then is surjective.

Proof. Let be fixed temporarily and the Yosida approximant of . For each , by using the inner product condition on and monotonicity of ( for all ), we see that for all for some Let Let be a real separable Hilbert space and a compact injection such that is dense in guaranteed by Lemma 4. Let be the natural injection and let and be adjoint of and , respectively. It follows that is a compact operator. Let First we show that is open in ; that is, is closed in To this end, let be a sequence in such that in as Since is continuously embedded in , we get in as Since is closed in , it follows that ; that is, This shows that is closed in ; that is, is open in The continuity of implies that is open in Since is continuously embedded in , it follows that where the closures are taken with respect to the spaces and , respectively. Since , we obtain that Since the sets and are disjoint, we conclude that For each , let be the Yosida approximant of Let It is known that, for each , is bounded, continuous, monotone, and of type . Let and be given by Since is continuous, it follows that is closed subset of We show that is a compact operator. To this end, let such that as Since is continuous from into , we have as Since , the sequence lies in . Since for all and , it follows that and for all Since and are demicontinuous, it follows that as By the density of in , it is known that is defined from into . As a result, for each , we see that for all . However, the right side expression goes to as ; that is, for each , it follows that On the other hand, by the density of in , for each , we get that is, as . Since is compact linear, which is completely continuous and , we arrive at as This shows that the mapping is continuous. Following similar argument as above, it is not difficult to show that maps any bounded subset of into relatively compact subset of . As a result, we conclude that is a compact operator. Fix In order to use Lemma 5, it is enough to show that (i) of Lemma 5 does not hold; that is, for all and , we have Suppose this is false; that is, there exist and such that . This yields We notice here that the continuity of , property of , and definition of boundary of an open set imply that holds. Since , it follows that By (11) and (20), we get which implies . But this is impossible because . Therefore, by applying Lemma 5, for each , , and , we conclude that the compact operator has a fixed point ; that is, Therefore, for each , there exists such that for all Since is bounded, the sequence is bounded. Since and are bounded, it follows that the sequence is bounded. Since and is continuously embedded, we see that , where the closures are with respect to the norms in and , respectively. As a result, the density of in implies that . By using (11), the monotonicity of and , and property of , we obtain that for all Since , it follows that for all . Consequently, we obtain that for all . Since is bounded and is quasibounded, we conclude that is bounded. Consequently, by using (24), it is not difficult to see that is bounded. If the sequence is bounded, then as . Otherwise, by using the boundedness of , we assume without loss of generality that as , , and as . Since , by choosing a sequence such that as and using (24) together with the monotonicity of , we get for all and Fixing and letting in (27), we obtain that Since is demicontinuous, letting , we arrive at that is, Since is generalized pseudomonotone, we conclude that , , and as For any , applying the monotonicity of , we arrive at Moreover, from (24), we obtain that for all . As a result, we arrive at From (31) and (33), we obtain for all . By the density of in and the continuity of , we conclude that for all Since, for any , for all , using in place of , we obtain that for all ; that is, for all Since is continuous and as , we have as . Letting , we arrive at for all Since is arbitrary, setting in place of yields for all Therefore, for each (by fixing temporarily), we see that there exists such that . Thus, for each , there exists such that for all . Since and are bounded, it follows that and are bounded. Since is quasibounded, it is not hard to see that is bounded, which implies the boundedness of . Assume without loss of generality that , and as . Since is generalized pseudomonotone with domain , it follows that Consequently, from (40), we arrive at Let be the Yosida resolvent of . It is well known that , , and for all Since and is bounded, it follows that as Thus, we have for all . Consequently, we have By the maximality of , applying Lemma 3, we obtain , , and as , which implies The generalized pseudomonotonicity of implies and As a result, letting in (40), we conclude that This implies that, for each , there exist and such that for all . Next we will show that is bounded. Assume without loss of generality that as . By the inner product condition on and monotonicity of with , we get for all ; that is, dividing this inequality by for all large , we get for all large . By using condition and (46), we get that for all . This gives for all . Consequently, the boundedness of follows. Since is quasibounded and , it is not hard to see that is bounded. Consequently, the boundedness of follows. Assuming that , , and as and using the arguments used in the first half of the proof of Theorem 6, we conclude that , , , and ; that is, for each , the inclusion problem is solvable in . Since is arbitrary, we obtain the surjectivity of . The proof using condition (ii) can be completed following similar arguments. The details are omitted here. This completes the proof.

It is worth mentioning that Theorem 6 is a new result because the perturbed operator is noncoercive and is densely defined such that contains a dense real separable reflexive Banach space. Under the conditions on , the result was unknown earlier even for coercive operator . The analog of Theorem 6 for single multivalued, finitely continuous, coercive, and quasibounded generalized pseudomonotone operator such that